Much ado about Mathieu

TL;DR

This paper proves that the elliptic genus of K3 surfaces exhibits a Mathieu Moonshine phenomenon, with class functions matching M24 characters, and explores the cohomological aspects and connections to other Moonshine phenomena.

Contribution

It establishes that the twisted elliptic genus functions are genuine M24 characters, confirming the Mathieu Moonshine conjecture and analyzing the role of group cohomology.

Findings

Twisted elliptic genus functions are true M24 characters.

Confirmed evenness of multiplicities in Mathieu Moonshine.

Identified cohomological interpretation of non-Fricke elements.

Abstract

Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II string theory on K3 surfaces appears to possess a Moonshine for the largest Mathieu group. Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24. In this paper we prove that the resulting sequence of class functions are true characters of M24, proving the Eguchi-Ooguri-Tachikawa conjecture. We prove the evenness property of the multiplicities, as conjectured by several authors. We also identify the role group cohomology plays in both K3-Mathieu Moonshine and Monstrous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton's Generalised Monstrous Moonshine conjecture. We investigate the intriguing proposal of Gaberdiel-Hohenegger-Volpato that K3-Mathieu Moonshine lifts to the Conway group Co1.